You can pretty much pack the same type of formal wear for Alaska as you would for a Caribbean cruise. Even for both, I pack a sweater for the chilly evenings and sometimes the dining room can have a chill.
You will pack totally different for the day-time however. Best to dress in layers and as the day warms up, you can take a sweat shirt off and or jacket. Pack a few t-shirts to wear under sweatshirts and a decent weight jacket. Unless you are hiking on a glacier, you won't need the super cold type jackets. Be sure to take something in case you encounter rain, most likely you will at some point. Good walking shoes and jeans or jogging type pants, whatever you are more comfortable wearing during the day for excursions. Also a hat and a pair of gloves will come in handy.
What ship are you on?